Expression of Concern: A white-box cryptographic SM2 scheme based on space hardness

The operating environments of a cryptographic algorithm have important impacts on its security. Traditional cryptographic algorithms need to run in a secure and trusted environment, in which case the internal state of these algorithms cannot be accessed by the outside. Namely, the cryptographic programs need to run under the black-box model. With the development of science and technology, the application scenarios of cryptographic algorithms are diversified, resulting in increasingly uncontrollable operating environments for cryptographic algorithms. White-box attack is a kind of terminal-based attack, which assumes that an adversary has complete control over the terminal. With this kind of attack, secret data can be easily exposed. In this paper, we propose a defense method based on strong (M,Z)- space hardness. The core idea is to put the secret value into the look-up table through some mathematical transformations. When the volume of the look-up table obtained by an adversary is limited, the secret value such as the private key cannot be recovered in polynomial time. Therefore, our proposed method is safe and efficient based on space hardness. We have designed a white-box implementation of SM2 digital signature algorithm to protect the private key under white-box attack. The main ideas presented in this paper can also be applied to constructing the white-box implementations of other asymmetric cryptographic algorithms.